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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
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GCE Ordinary Level
MARK SCHEME for the May/June 2009 question paper
for the guidance of teachers
4037 ADDITIONAL MATHEMATICS
4037/01
Paper 1, maximum raw mark 80
This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of
the examination. It shows the basis on which Examiners were instructed to award marks. It does not
indicate the details of the discussions that took place at an Examiners’ meeting before marking began,
which would have considered the acceptability of alternative answers.
Mark schemes must be read in conjunction with the question papers and the report on the
examination.
•
CIE will not enter into discussions or correspondence in connection with these mark schemes.
CIE is publishing the mark schemes for the May/June 2009 question papers for most IGCSE, GCE
Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
Page 2
Mark Scheme: Teachers’ version
GCE O LEVEL – May/June 2009
Syllabus
4037
Paper
01
Mark Scheme Notes
Marks are of the following three types:
M
Method mark, awarded for a valid method applied to the problem. Method marks are
not lost for numerical errors, algebraic slips or errors in units. However, it is not
usually sufficient for a candidate just to indicate an intention of using some method or
just to quote a formula; the formula or idea must be applied to the specific problem in
hand, e.g. by substituting the relevant quantities into the formula. Correct application
of a formula without the formula being quoted obviously earns the M mark and in some
cases an M mark can be implied from a correct answer.
A
Accuracy mark, awarded for a correct answer or intermediate step correctly obtained.
Accuracy marks cannot be given unless the associated method mark is earned (or
implied).
B
Accuracy mark for a correct result or statement independent of method marks.
•
When a part of a question has two or more “method” steps, the M marks are generally
independent unless the scheme specifically says otherwise; and similarly when there are
several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a
particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme.
When two or more steps are run together by the candidate, the earlier marks are implied and
full credit is given.
•
The symbol √ implies that the A or B mark indicated is allowed for work correctly following
on from previously incorrect results. Otherwise, A or B marks are given for correct work
only. A and B marks are not given for fortuitously “correct” answers or results obtained from
incorrect working.
•
Note:
B2 or A2 means that the candidate can earn 2 or 0.
B2, 1, 0 means that the candidate can earn anything from 0 to 2.
© UCLES 2009
Page 3
Mark Scheme: Teachers’ version
GCE O LEVEL – May/June 2009
Syllabus
4037
Paper
01
The following abbreviations may be used in a mark scheme or used on the scripts:
AG
Answer Given on the question paper (so extra checking is needed to ensure that
the detailed working leading to the result is valid)
BOD
Benefit of Doubt (allowed when the validity of a solution may not be absolutely
clear)
CAO
Correct Answer Only (emphasising that no “follow through” from a previous error
is allowed)
ISW
Ignore Subsequent Working
MR
Misread
PA
Premature Approximation (resulting in basically correct work that is insufficiently
accurate)
SOS
See Other Solution (the candidate makes a better attempt at the same question)
Penalties
MR –1
A penalty of MR –1 is deducted from A or B marks when the data of a question or
part question are genuinely misread and the object and difficulty of the question
remain unaltered. In this case all A and B marks then become “follow through √ ”
marks. MR is not applied when the candidate misreads his own figures – this is
regarded as an error in accuracy.
OW –1,2 This is deducted from A or B marks when essential working is omitted.
PA –1
This is deducted from A or B marks in the case of premature approximation.
S –1
Occasionally used for persistent slackness – usually discussed at a meeting.
EX –1
Applied to A or B marks when extra solutions are offered to a particular equation.
Again, this is usually discussed at the meeting.
© UCLES 2009
Page 4
1
Mark Scheme: Teachers’ version
GCE O LEVEL – May/June 2009
12 = 15θ , θ = 0.8 rads
1
(ii) Area = 152 ( 0.8 )
2
(i)
leading to 90 (cm2)
Syllabus
4037
M1, A1 [2]
M1 for use of s = rθ
M1
M1 for use of A =
Paper
01
1 2
rθ
2
A1
[2]
2
x3 = 8, leading to x = 2
B1
B1 for finding where curve crosses the x axis
1
dy
= 3 x 2 leading to grad of −
12
dx
M1
M1 for attempt to differentiate and use of
m1m2 = –1
DM1
A1
DM1 for attempt at equation of normal
Allow unsimplified
for normal
1
(x − 2)
12
1
1

y =− x+ 
12
6

y−0 = −
[4]
3
1 − cos 2 θ sin 2 θ
=
sec 2 θ − 1 tan 2 θ
= cos 2 θ
= 1 − sin 2 θ
M1
M1
M1 for use of 1 − cos 2 θ = sin 2 θ
M1 for use of sec 2 θ − 1 = tan 2 θ
M1
A1
M1 for attempt to simplify
[4]
Alt Scheme
1 − cos 2 θ
sin 2 θ
=
sec 2 θ − 1 1 − cos 2 θ
cos 2 θ
sin 2 θ cos 2 θ
=
sin 2 θ
= cos 2 θ
= 1 − sin 2 θ
4
(i)
5 x − 3 = kx 2 − 3 x + 5
kx 2 − 8 x + 8 = 0
using b 2 − 4ac = 0 , k = 2
(Alt scheme: 5 = 2kx − 3 , x =
20
16 12
−3 = − +5
k
k k
leading to k = 2 )
(ii) leading to x = 2, y = 7
M1
M1 for use of 1 − cos 2 θ = sin 2 θ
M1
M1 for attempting to get all in terms of cos
M1
M1 for attempt to simplify
A1
M1
M1 for equating line and curve equations
DM1, A1
[3]
DM1 for use of b 2 − 4ac on resulting
quadratic
4
k
(Alt scheme: M1 for attempt to differentiate
quadratic and equate to 5
DM1 for simplification and solution using
resulting quadratic
M1 for obtaining x and y coords
M1, A1
[2]
© UCLES 2009
Page 5
5
Mark Scheme: Teachers’ version
GCE O LEVEL – May/June 2009
2( 2 x −1)
= 33 x
4 x − 2 = 3x
x=2
(a) 3
Syllabus
4037
Paper
01
2( 2 x−1)
B1
B1 for 3
B1
B1 for 33 x
B1
B1 for x = 2
[3]
b
(allow here)
a2
p = −2, q = 1
B1
B1
f (3), f (−5) or f (0.5) = 0 spotted
B1
B1 for spotting one root
M1
M1 for attempt to obtain quadratic factor
A1
A1 all correct
M1
A2,1,0
M1 for solution of quadratic
A2 for all 3 solutions (–1 each error)
Correct factors only – lose 1 A mark
(b) a −2b or
6
(2 x − 1)(x + 2 x − 15)
(x + 5)(2 x 2 − 7 x + 3)
(x − 3)(2 x 2 + 9 x − 5)
2
Either
Or
Or
x = 3, –5, 0.5
B1 for each
[2]
[6]
7
(i)
(ii)
3 xe3 x + e3 x − e3 x
= 3 xe3 x
M1, A1, B1
1  3 x e3 x 
3x
e
d
x
x
=
 xe −

∫
3
3 
DM1
DM1
A1
M1 for attempt to differentiate a product.
A1 for correct product.
[3] B1 for −e3 x
DM1 for recognition of the ‘reverse’ to (i)
DM1 for dealing with ‘3’
A1 all correct (condone omission of c)
[3]
8
(i)
(
)
x 2 + 9 2 − 2 x(2 x )
dy
=
2
dx
x2 + 9
=
(
18 − 2 x 2
(x
2
+9
x= ±3
)
2
)
, turning points,
B2,1,0
Attempt to differentiate a quotient
–1 each error
M1
M1 for correct attempt to find the turning
points.
A1 for both
A1
[4]
(ii)
dx
=2
dt
dy
 16 
= 2×

dt
 100 
8
= 0.32 or
25
dx
=2
dt
B1
B1 for use of
M1
M1 for use of rates of change
A1
[3]
© UCLES 2009
Page 6
9
(i)
Mark Scheme: Teachers’ version
GCE O LEVEL – May/June 2009
1 
 1
i+
j  = 10i + 10 j
10 2 
2 
 2
Syllabus
4037
Paper
01
M1 for attempt at a correct direction vector
A1 all correct
M1
A1
[2]
(ii)
( −4i + 8 j) + ( 20i + 20 j) = 16i + 28 j
M1
A1
(iii) (10i + 10 j) − ( 8i + 6 j) = 2i + 4 j
M1
A1
M1 for valid attempt
A1 all correct
[2]
M1 for attempt at vector difference
A1 condone negative
[2]
(iv) displacement of
(19i + 34 j) − (16i + 28 j) = 3i + 6 j
M1
time =1330 hours
(accept 1.5 hours)
at 31i + 43 j
A1
M1 for displacement and attempt to obtain
time
A1 for correct time
A1
A1 for correct position vector
[3]
Alternative scheme:
(19i + 34 j) + ( 8i + 6 j) t =
(16i + 28 j) + (10i + 10 j) t
M1 for attempt to equate like vectors
A marks as above
or equivalent
10 (i) mAB = 0.75
line AB y − 0 = 0.75( x + 4)
M1
A1
M1 for attempt at mAB and line AB
4
3
M1
M1 for use of ‘m1m2 = –1’ and attempt at
line PQ
mPQ = −
line PQ y − 10 = −
4
( x − 1)
3
intersection at C (4, 6)
Q (8.5 0)
A1
M1
A1
√ B1
M1 for attempt at solving simultaneous
equations
Ft on their line PQ
[7]
(ii) AC = 10, CQ = 7.5
Area = 37.5
M1
A1
M1 for attempt at lengths and area
[2]
© UCLES 2009
Page 7
Mark Scheme: Teachers’ version
GCE O LEVEL – May/June 2009
11 (i) ln s
ln t
1.6
ln s
7.2
Plot
= n ln t + ln k
2.7
5.9
3.4 4.2
5
4
ln s against ln t
Paper
01
M1for attempt to take logs
A1 for correct form
M1 for attempt to plot correct graph
A1 for a reasonable straight line
M1, A1
M1
A1
4.6
3.6
Syllabus
4037
[4]
(ii) grad n = –1.2 (–1.4 to –1.0)
Intercept = ln k, leading to
k = 7900 – 10 000
M1, A1
M1, A1
(iii) when t = 50, ln t = 4.4
leading to s = 80 (72 – 92)
M1
A1
M1 for use of grad = n
M1 for use of intercept = ln k
[4]
M1 for attempt to obtain s
[2]
Alternative method
(i) lg s = n lg t + lg k
lg t
0.7 1.2 1.5 1.8
lg s
3.1 2.5 2.2 1.7
10
2
1.6
Same scheme applies
ln s
9
8
7
6
5
4
3
2
1
ln t
1
2
© UCLES 2009
3
4
5
Page 8
Mark Scheme: Teachers’ version
GCE O LEVEL – May/June 2009
12 EITHER
(i) amplitude = 1
(ii) period = 6π, 18.8
 x
3
(iii) sin   =
1
π 5π
, x= ,
2 2
2
B1
[1]
B1
[1]
Syllabus
4037
Paper
01
M1 for attempt to solve correctly
A1 for each (allow degrees here)
M1
A1, A1
[3]
(iv) Area under curve
5π
2
5π
x 2
x


∫π 1 + sin 3 dx =  x − 3 cos 3  π
M1
M1 for attempt to integrate
B1, B1
B1 for x, B1 for −3cos
DM1
DM1 for correct use of limits
M1
M1 for attempt at rectangle plus subtraction –
must be working in radians
2
2
leading to 2π + 3 3
 5π
−
 2
Area of rectangle = 
π 3
×
2 2
= 3π
Shaded area = 3 3 – π (2.05)
x
3
A1
[6]
Alternative solution: Shaded area
5π
2
5π
x 2
x



=
−
0.5
x
−
3cos
sin
0
.
5
d
x
−


∫π  3

3  π

2
2
M1
M1
M1 for subtraction (must be using radians)
M1 for attempt to integrate
B1, B1
B1 for −0.5x , B1 for −3cos
DM1, A1
DM1 for correct use of limits
© UCLES 2009
x
3
Page 9
OR
(i)
t=
Mark Scheme: Teachers’ version
GCE O LEVEL – May/June 2009
π
8
Syllabus
4037
Paper
01
B1
[1]
(ii) a = −4k sin 4t
M1, A1
M1 for attempt to differentiate
[2]
3π
(iii) 12 = −4k sin
leading to
2
k =3
M1 for attempt to substitute into their
acceleration equation
M1
A1
[2]
(iv)
4 v
B1
B1 for correct shape
√ B1
B1 ft on their value for k
3
2
1
t
1
2
3
–1
–2
[2]
–3
–4
(v) s =
π
24
∫ 3 cos 4t.dt
M1, √ A1
M1 for attempt to integrate
Ft on their value for k
DM1, A1
[4]
DM1 for application of limits or equivalent
0
π
3
3
 24
=  sin 4t  leading to
8
4
0
© UCLES 2009